Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in three-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite nonovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family $\mathcal{ℱ}$ in one of the known finite circle geometries of order $q$, with $\mid \mathcal{ℱ}\mid \ge \frac{1}{\sqrt{2}}{q}^2+2\sqrt{2}q+8$, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.

中文翻译：

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圆几何中 Erdős–Ko–Rado 定理的稳定性

圆几何是在三维空间中捕捉球体、圆锥和双曲面上的圆几何的入射结构。在之前的一篇论文中，作者描述了有限卵形圆几何中最大的相交族，除了奇数阶的莫比乌斯平面。在本文中，我们展示了在这些莫比乌斯平面中，如果阶数大于 3，则最大的相交族是通过固定点的圆的集合。我们在唯一已知的有限非圆形几何形状家族中展示了相同的结果。使用相同的技术，我们展示了在所有卵圆形几何形状中的大相交族的稳定性结果。更具体地说，我们证明了一个相交的家庭$\mathcal{ℱ}$在已知的有限圆有序几何之一中$q$， 和$\mid \mathcal{ℱ}\mid \ge \frac{1}{\sqrt{2}}{q}^2+2\sqrt{2}q+8$, 必须由通过一个公共点的圆组成，或者在偶数阶的拉盖尔平面的情况下通过一个公共核。